# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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y = (x- 3)(s + 1)

0 = (x β 3)(x + 1) Substitute y = 0 to obtain the x intercepts x = 3 and x = β 1 are the x intercepts Solving the quadratic equation

The y intercept is the point where the curve crosses the y-axis. To obtain this point we substitute x = 0 into the equation of the curve and find the value of y:

y = (x β 3)(x + 1)

y = (0 β 3)(0 +1) Substitutes = 0 to obtain the y intercept

y = -3 x 1

y = β 3 is the y intercept

We can now sketch the parabola. We know it is βcupβ-shaped and crosses the y-axis at β3 and the s-axis at the two points 3 and β 1. The axis of symmetry is the vertical line that passes through the vertex, and is drawn dashed in figure f. It crosses the s-axis at x = 1, which is halfway between the x intercepts 3 and β 1. We now substitute the value x = 1 into the equation of the curve to obtain the y coordinate

366 QUADRATIC GRAPHS

of the vertex:

y = (1 β 3)(1 + 1) Substituting x = 1 into the equation of the curve y = β2x2

y = -4

The coordinates of the vertex are x = 1 and y = β4, or more briefly (1, β4).

Sketching Cubic Graphs Cubic equations are of the form y = ax3 + bx2 + cx + d, where b, c, and d may be zero. Under this entry we will study the graphs of cubics in factorized form. Typical examples of cubics in factorized form are

y = (x - 2)(x + 2)(x + 3)

y = 2x{x β 3)(x + 4)

y = (x - 2)(x + 3)2

y = (3 β x)(x + 2)(x β 4)

These types of cubics have two basic shapes, depending on whether or not the value of a'my = ax3 + bx2 + cx + d is positive or negative (see figure g). A cubic graph is not made up of two parabolas joined together, but has its own characteristics.

a > 0 a < 0

(g)

Example 4. Sketch the graph of the cubic y = (x β 2)(x + 2)(x + 4).

Solution. When the brackets are expanded the value of a is +1, and so its basic shape is of the type in which a > 0. The x intercepts are found by substituting y = 0

QUADRILATERAL 367

into the equation of the curve:

0 = (x β 2)(x + 2)(x + 4) Substituting y = 0

x = 2, x = β 2, a: = ββ 4 Solving the equation

The y intercept is found by substituting x = 0 into the equation of the curve:

y = (0 β 2)(0 + 2)(0 + 4) Substituting x = 0

y = β2x2x4 y = -16

The graph of y = (x β 2)(x + 2)(x + 4) will cross the x-axis at x = 2, x = β 2, a: = β4 and it will cross the y-axis at y = β16 (see figure h). Since the y-axis needs to go down rather a long way, to y = β16, it is a good idea to reduce the scale on this axis.

Without further work, which is beyond the scope of this book, we do not know the coordinates of the turning points, but endeavor to draw a smooth curve through the intercepts.

References: Factorize, Intercept, Parabola, Substitution, Translation, Turning Points, Vertex.

QUADRILATERAL

A quadrilateral is a polygon with four sides. See the entry Polygon for the angle sum of a quadrilateral. Many of the following quadrilaterals are studied separately under their own names:

β¦ Square (this is a regular quadrilateral)

β¦ Rectangle

368 QUALITATIVE DATA

β¦ Rhombus

β¦ Parallelogram

β¦ Kite

β¦ Trapezium

Figure a shows some quadrilaterals; the last one is a regular quadrilateral, or square.

Quadrilaterals will always tessellate. Suppose we draw any quadrilateral, like the one shaded in figure b. The white tile is obtained by rotating the shaded tile through half a turn, which is 180Β°. This process is repeated to obtain the tiling pattern drawn here.

References: Mosaics, Polygon, Regular Polygon, Rotation, Tessellations, Triangle, Tiling Patterns.

QUADRUPLE

This word means βfour timesβ. For example, suppose John invested $100 and in 5 years he quadrupled his investment. This means that his investment increased four times to the value of 4 x $100 = $400.

QUALITATIVE DATA

(a)

(b)

Reference: Data.

QUESTIONNAIRE 369

QUALITATIVE GRAPHS

The data that are used in most graphs are quantitative data about numbers and relationships between them. Qualitative data are descriptive data, such as eye color, gender, nationality, attitudes, and so on. When we study qualitative graphs we are examining relationships between items of descriptive dataβparticularly attitudes. For example, the figure shows the attitudes of four students towards their study of English and Mathematics. Boz dislikes English and has a fairly good attitude to Math. Ray hates both subjects. Liz likes English more than Math. Ron likes Math and quite likes English.

Loves English

Hates

Math

β’ Liz

Ron β’

.Ray 'Boz

Loves

Math

Hates English

References: Data, Graphs, Qualitative Data, Quantitative Data.

QUANTITATIVE DATA

Reference: Data.

QUARTILES

References: Box and Whisker Graph, Interquartile Range.

QUESTIONNAIRE

A survey is an inquiry into something, such as how shoppers feel about alcohol being sold in supermarkets. A survey may take the form of a questionnaire to be completed by a sample of the population. The information gathered from the sample is used to make predictions and draw conclusions about the whole population. For example, it may be possible from the responses of the questionnaire to conclude that it is not popular with people generally to sell alcohol in supermarkets. The questionnaire will be designed to gather information that truly represents the attitude of the people toward the sale of alcohol, and not to produce a biased view to support the people who own wine shops! The questionnaire is a set of written questions, and is not the same as a verbal interview.

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